Circular numerical ranges, Blaschke products, and Poncelet curves: Why n circles are better than one

Abstract

Suppose that a circle is inscribed in a convex polygon that is inscribed in the unit circle. When we draw all possible lines between distinct vertices, these will be tangent to a package of circles. If we know where the center of one of these circles is, can we tell where all the others are? We investigate this question when one of the centers is at 0, provide two geometric proofs for this, and use it to provide new proofs of some recent results on Blaschke products and the numerical ranges of partial isometries. The latter result concerns lower-dimensional cases of a matrix theoretic conjecture of Gau, Wang, and Wu, which is still open in general.